Question

Evaluate \( \int_0^{\frac{\pi}{2}} \left( e^{\sin x} - e^{\cos x} \right) \, dx \)

• A. \( \frac{1}{2} \)
• B. 0
• C. 1
• D. \( \frac{\pi}{4} \)

Hint:

When evaluating integrals of periodic functions over symmetric intervals, check for symmetry and periodicity to simplify calculations.

Answer 1

The Correct Option is B

Step 1: Analyze the integrals.
The integral involves the difference of two functions, \( e^{\sin x} \) and \( e^{\cos x} \). Both functions are periodic with a period of \( \pi \), and over the interval \( \left[ 0, \frac{\pi}{2} \right] \), their values cancel out.
Step 2: Conclusion.
The integral evaluates to 0. Hence, the correct answer is (B) 0.